Shadow Douglas–Rachford Splitting for Monotone Inclusions

Csetnek, Ernö Robert; Malitsky, Yura; Tam, Matthew K.

doi:10.1007/s00245-019-09597-8

Cited by 51 publications

(36 citation statements)

References 24 publications

(44 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[23]. If we discretize Heavy Anchor and restrict the parameter values, we can recover the optimistic gradient-descent/ascent (OGDA) [24] or the shadow Douglas Rachford [25]. However, all these optimization methods assume that the map is the gradient of a convex function.…”

Section: Introductionmentioning

confidence: 99%

On the exact convergence to Nash equilibrium in monotone regimes under partial-information

Gadjov

Pavel

2020

2020 59th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

In this paper, we consider distributed Nash equilibrium seeking in monotone and hypomonotone games. We first assume that each player has knowledge of the opponents' decisions and propose a passivity-based modification of the standard gradient-play dynamics, that we call "Heavy Anchor". We prove that Heavy Anchor allows a relaxation of strict monotonicity of the pseudo-gradient, needed for gradient-play dynamics, and can ensure exact asymptotic convergence in merely monotone regimes. We extend these results to the setting where each player has only partial information of the opponents' decisions. Each player maintains a local decision variable and an auxiliary state estimate, and communicates with their neighbours to learn the opponents' actions. We modify Heavy Anchor via a distributed Laplacian feedback and show how we can exploit equilibrium-independent passivity properties to achieve convergence to a Nash equilibrium in hypomonotone regimes. N j=1 w ij . Assume that W = W T so the weighted Laplacian of G is L = Deg − W . When G is connected and undirected, 0 is a simple eigenvalue of L, L1 N = 0, 1 T N L = 0 T , and all other eigenvalues are positive, 0 < λ 2 (L) ≤ • • • ≤ λ N (L).

show abstract

Section: Introductionmentioning

confidence: 99%

On the exact convergence to Nash equilibrium in monotone regimes under partial-information

Gadjov

Pavel

2020

2020 59th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

show abstract

“…We want also to mention [6,7], where continuous and discrete versions of the same algorithm are combined with Tikhonov regularization terms, and [8,9], where second order dynamical systems of Nesterov type with Hessian driven damping and corresponding inertial proximal point type algorithms are investigated. In the context of monotone inclusions we refer to [10] for a so-called shadow Douglas-Rachford splitting in both continuous and discrete versions, and to [11] for a forward-backward-forward di erential equation and its discrete counterpart.…”

Section: Introductionmentioning

confidence: 99%

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

Boţ

Grad

Meier

et al. 2020

Advances in Nonlinear Analysis

View full text Add to dashboard Cite

In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

show abstract

“…Its name originates from online optimization [53,54]. The idea dates back even further [52] and has been studied in the mathematical programming community as well [14,41,43].…”

Section: Related Literaturementioning

confidence: 99%

Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions

Böhm¹

2022

Preprint

View full text Add to dashboard Cite

We investigate a structured class of nonconvex-nonconcave min-max problems exhibiting so-called weak Minty solutions, a notion which was only recently introduced, but is able to simultaneously capture different generalizations of monotonicity. We prove novel convergence results for a generalized version of the optimistic gradient method (OGDA) in this setting matching the ones recently shown for the extragradient method (EG). In addition we propose an adaptive stepsize version of EG, which does not require knowledge of the problem parameters.

show abstract

Shadow Douglas–Rachford Splitting for Monotone Inclusions

Cited by 51 publications

References 24 publications

On the exact convergence to Nash equilibrium in monotone regimes under partial-information

On the exact convergence to Nash equilibrium in monotone regimes under partial-information

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions

Contact Info

Product

Resources

About